Because this is a cylinder, everything is "linear" and you don't need Calculus. It is easy to find the center of gravity of the water. Taking a coordinate system with origin at the center of the bottom of the pool, and the positive z-axis upward, and units of feet, the center of gravity of the water is at (0, 0, 1.5). That means that the water, volume cubic feet and so (235.6)(65)= 15315 pounds must be lifted an average of 11,5- 1.5= 10 feet. The work necessary is 15315(10)= 153150 foot-pounds.
Your error, I believe, was multiplying by 9.8. That is the acceleration due to gravity in the metric system and multiplying by it will convert mass to weight in the metric system. But here, you were given the weight density of water, not mass density, as pounds per cubic foot, in the English system.